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If you have Medicare coverage or you help someone who does, it’s time for your annual homework assignment: comparing your options during open enrollment to see if you can do better.

It can be intimidating, but the payoff for your effort is potential savings of hundreds to thousands of dollars.

Through Dec. 7, Americans enrolled in Medicare, the federal health-insurance program for people 65 and older and the disabled, can make changes to how they receive their benefits. Those changes will take effect Jan. 1.

In the next few weeks, beneficiaries can switch their stand-alone Part D prescription-drug plan or enroll in one for the first time, if they didn’t sign up when they were first eligible. They also can join a Medicare Advantage plan, a private health plan that wraps medical and often drug benefits into a single HMO- or PPO-like product. Or they can drop out of a Medicare Advantage plan in favor of original Medicare, the government-sponsored program that doesn’t have restricted networks of doctors and hospitals.

“What is daunting to Medicare beneficiaries is the sheer number of private-plan options out there,” said Fred Riccardi, director of programs and outreach for the Medicare Rights Center, a nonprofit group in New York. “They all have different premiums and co-payments and different covered drugs and restrictions on drugs.”

If you need help weighing your options or enrolling in a Medicare plan, there are several free resources you can consult.

-- Medicare.gov has a Plan Finder tool that works by plugging in your ZIP code. Counselors also can assist you by calling 800-Medicare (800-633-4227.)

-- You also can call the companies that provide the plans for help understanding their offerings. The Kaiser Family Foundation offers descriptions of how Medicare works on its website, KFF.org.

Here are five of the costliest mistakes beneficiaries make during open enrollment, according to experts:

1: Not bothering to give your current coverage a checkup.

The first step is to take stock of what you have. Even if you like your drug coverage, make sure you review your plan’s annual notice of change, a letter from the companies that Part D enrollees should have received by now, Riccardi said.

“It should say whether a drug that they’re taking is no longer covered,” he said. “It should specify how their premium is changing or if a pharmacy is leaving the network.”

If you don’t have any drug coverage, now is the time to consider hopping on board. Should you develop a health condition that requires prescription medication, your out-of-pocket costs without insurance could easily overshadow the 1 percent per month late-enrollment penalty you may accrue for delaying enrollment in Medicare Part D.

2: Failing to shop around or only considering premium costs if you do.

If you pass on seeing what’s out there, you run the risk of overpaying for your drugs next year by default.

“You really have to look to see that the drugs (you take) are covered at the best possible price,” said Katy Votava, founder and president of Goodcare.com, a consulting service focused on Medicare and health-care costs. At minimum, beneficiaries should compare drug plans every two years because costs often creep up, she said. “People assume drug coverage is more standardized than it is.”

In fact, drug plans, which you can compare at Medicare.gov, are all over the map in terms of how they structure premiums, deductibles, co-payments and tiers of coverage.

“You could have a $10-a-month co-pay vs. a $40-a-month co-pay. You could have one medication covered for $30 a month and the other not covered at all,” Votava said, noting it’s possible to save $2,000 to $4,000 just by being in the right plan.

In addition to making sure the drugs you need are on the formulary, or list of covered drugs, and that your pharmacy is in the network, see whether plans impose prior authorization, step therapy, quantity limits or other restrictions, Riccardi said. Reasons to shop around include if your medications have cost you a lot in the last year, if premiums are increasing to an uncomfortable level, if you’re using a lot of out-of-network care in a Medicare Advantage plan, or if you’ve experienced poor customer service.

It’s also easier to find quality plans this year, according to the Centers for Medicare & Medicaid Services, which has beefed up its star ratings system to alert consumers to the best-performing plans and remind those stuck in continuously low-performing ones that they can switch plans. Beneficiaries have 127 four-star or five-star Medicare Advantage plans from which to choose, up from 106 during open enrollment for 2012. And those in original Medicare have 26 high-performing prescription-drug plans at their disposal, up from 13 last year.

3: Failing to account for out-of-pocket maximums.

This is the number that tells you how much you could pay in a year before the plan kicks in to cover what’s considered catastrophic costs. Read the fine print to understand what the plan does and doesn’t count toward its out-of-pocket max. Factoring in worst-case scenarios could save you thousands of dollars if you develop a condition that requires extensive health services or a pricey prescription drug. Then compare plans and do the math.

“If you’ve had a Medicare Advantage plan where all your care providers were in the network and it had a low out-of-pocket max, like $1,700, then most people couldn’t get a Medigap (supplemental insurance) plan that would cost less than that,” Votava said. Otherwise, going with original Medicare and a Medigap plan to cover out-of-pocket costs might make more sense.

4: Choosing a Medicare Advantage plan without first checking if your doctors are in network.

About 1 in 4 Medicare beneficiaries chooses a Medicare Advantage plan, which sometimes offers benefits beyond what’s included in traditional Medicare. If you’re considering a Medicare Advantage plan, remember that this model means seeing out-of-network providers can quickly become a costly proposition for you. Before signing up for this option, call your preferred doctors, specialists and hospitals to verify that they participate in the plan’s network. This can get tricky if you travel a lot, spend winters in a different location, or get a referral from your primary-care doctor to a specialist who’s out of network.

The good news is that while plans can add or subtract health-care providers from their networks every year, they can’t be so picky about their members, said David Lipschutz, policy attorney for the Center for Medicare Advocacy in Washington. “Medicare Advantage plans have to take all comers, with the minor caveat” of people with end-stage renal disease.

5: Assuming retiree health coverage from a former employer is automatically the best deal or misunderstanding how it interacts with Medicare’s various parts.

Retirees are often loyal to their old employers, Votava said, but their retiree plan may not be the gold standard in terms of value for their money. In some cases, retirees could get better coverage at a lower cost by going with original Medicare and a Medigap plan or a Medicare Advantage plan. A basic rule of thumb is if seniors are spending more than $250 to $300 a month for their retiree coverage, they should shop around, Votava said. “Even people who are paying $200 could be paying $125.”

Still, the decision of whether to drop retiree coverage can be complex, and it’s often irreversible, so take your time and seek professional advice if you need it.

If you have retiree coverage from a former employer, make sure you follow its rules concerning Medicare Advantage and Part D, Lipschutz said. Some retiree plans require you to enroll in a Medicare Advantage plan. Some will work with Part D plans while others prohibit you from signing up for a stand-alone drug plan, he said.

“Sometimes people will enroll in a Medicare Part D plan and that enrollment might jeopardize their retiree coverage all together,” Lipschutz said.